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module_bl_keps.F
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MODULE module_bl_keps
real vk,temin,dmin,tpemin,c1,c2,c3,c_mu,Pr ! constant from Zonato et al.,2022
parameter(temin=1E-4,dmin=1E-7,tpemin=1E-7,c_theta=0.09, &
vk=0.4,c_mu=0.09,c1=1.44,c2=1.92,c3=1.44,c4=0.27,c5=0.08,sigma_eps=1.3) ! impose minimum values for tke similar to those of MYJ
parameter(temax=50.,dmax=50.,tpemax=5.)
integer,private,parameter:: bl_keps_adv = 0
! -----------------------------------------------------------------------
CONTAINS
subroutine keps(mol,tsk,xtime,frc_urb2d,flag_bep,dz8w,dt,u_phy,v_phy,th_phy &
,PI_PHY,RTHRATEN,P8W &
,rho,qv_curr,qc_curr,hfx &
,qfx,ustar,cp,g &
,rublten,rvblten,rthblten &
,rqvblten,rqcblten &
,tke_pbl,diss_pbl,tpe_pbl &
,tke_adv,diss_adv,tpe_adv &
,pr_pbl &
,wu,wv,wt,wq,exch_h,exch_m,pblh &
,a_u_bep,a_v_bep,a_t_bep,a_q_bep &
,b_u_bep,b_v_bep &
,b_t_bep,b_q_bep &
,b_e_bep &
,sf_bep,vl_bep &
,br,znt,psim,psih &
,ids,ide, jds,jde, kds,kde &
,ims,ime, jms,jme, kms,kme &
,its,ite, jts,jte, kts,kte)
implicit none
!-----------------------------------------------------------------------
! Input
!------------------------------------------------------------------------
INTEGER:: ids,ide, jds,jde, kds,kde, &
ims,ime, jms,jme, kms,kme, &
its,ite, jts,jte, kts,kte
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) :: DZ8W !vertical resolution
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) :: qv_curr !moisture
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) :: qc_curr !liquid water
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) :: RHO !air density
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) :: TH_PHY !potential temperature
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) :: PI_PHY
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) :: RTHRATEN
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) :: P8W
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) :: U_PHY !x-component of wind
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) :: V_PHY !y-component of wind
REAL, DIMENSION( ims:ime, jms:jme ), INTENT(IN ) :: ustar !friction velocity
REAL, DIMENSION( ims:ime, jms:jme ), INTENT(IN ) :: hfx !sensible heat flux (W/m2) at surface
REAL, DIMENSION( ims:ime, jms:jme ), INTENT(IN ) :: qfx !moisture flux at surface
REAL, DIMENSION( ims:ime, jms:jme ), INTENT(IN ) :: tsk,mol !moisture flux at surface
real, INTENT(IN ) :: g,cp !gravity and Cp
REAL, INTENT(IN ):: DT ! Time step
REAL, DIMENSION( ims:ime, jms:jme ), INTENT(IN ) :: br
REAL, DIMENSION( ims:ime, jms:jme ), INTENT(IN ) :: znt
REAL, DIMENSION( ims:ime, jms:jme ), INTENT(IN ) :: FRC_URB2D !fraction cover urban
REAL, DIMENSION( ims:ime, jms:jme ), INTENT(INOUT) :: PBLH !PBL height
REAL, DIMENSION( ims:ime, jms:jme ), INTENT(IN ) :: psim,psih
! variable added for urban
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) ::a_u_bep ! Implicit component for the momemtum in X-direction
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) ::a_v_bep ! Implicit component for the momemtum in Y-direction
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) ::a_t_bep ! Implicit component for the Pot. Temp.
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) ::a_q_bep ! Implicit component for Moisture
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) ::b_u_bep ! Explicit component for the momemtum in X-direction
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) ::b_v_bep ! Explicit component for the momemtum in Y-direction
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) ::b_t_bep ! Explicit component for the Pot. Temp.
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) ::b_q_bep ! Explicit component for Moisture
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) ::b_e_bep ! Explicit component for Moisture
! urban surface and volumes
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) ::sf_bep ! surface of the urban grid cells
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(IN ) ::vl_bep ! volume of the urban grid cells
LOGICAL, INTENT(IN) :: flag_bep !flag for BEP
!
!-----------------------------------------------------------------------
! Local, carried on from one timestep to the other
!------------------------------------------------------------------------
! real, save, allocatable, dimension (:,:,:)::TKE ! Turbulent kinetic energy
real, dimension (ims:ime, kms:kme, jms:jme) ::th_0 ! reference state for potential temperature
!------------------------------------------------------------------------
! Output
!------------------------------------------------------------------------
real, dimension( ims:ime, kms:kme, jms:jme ), INTENT(OUT ) :: exch_h ! exchange coefficient for heat
real, dimension( ims:ime, kms:kme, jms:jme ), INTENT(OUT ) :: exch_m ! exchange coefficient for momentum
real, dimension( ims:ime, kms:kme, jms:jme ), INTENT(INOUT ) :: tke_pbl ! Turbulence Kinetic Energy
real, dimension( ims:ime, kms:kme, jms:jme ), INTENT(INOUT ) :: diss_pbl ! Dissipation Rate
real, dimension( ims:ime, kms:kme, jms:jme ), INTENT(INOUT ) :: tpe_pbl ! Dissipation Rate
real, dimension( ims:ime, kms:kme, jms:jme ), INTENT(INOUT ) :: pr_pbl ! Dissipation Rate
real, dimension( ims:ime, kms:kme, jms:jme ), INTENT(INOUT ) :: tke_adv ! Dissipation Rate
real, dimension( ims:ime, kms:kme, jms:jme ), INTENT(INOUT ) :: diss_adv ! Dissipation Rate
real, dimension( ims:ime, kms:kme, jms:jme ), INTENT(INOUT ) :: tpe_adv ! Dissipation Rate
real, dimension( ims:ime, kms:kme, jms:jme ), INTENT(OUT ) :: wu ! Turbulent flux of momentum (x)
real, dimension( ims:ime, kms:kme, jms:jme ), INTENT(OUT ) :: wv ! Turbulent flux of momentum (y)
real, dimension( ims:ime, kms:kme, jms:jme ), INTENT(OUT ) :: wt ! Turbulent flux of temperature
real, dimension( ims:ime, kms:kme, jms:jme ), INTENT(OUT ) :: wq ! Turbulent flux of water vapor
! only if idiff not equal 1:
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(OUT ) :: RUBLTEN !tendency for U_phy
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(OUT ) :: RVBLTEN !tendency for V_phy
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(OUT ) :: RTHBLTEN !tendency for TH_phy
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(OUT ) :: RQVBLTEN !tendency for QV_curr
REAL, DIMENSION( ims:ime, kms:kme, jms:jme ), INTENT(OUT ) :: RQCBLTEN !tendency for QV_curr
!--------------------------------------------------------------
! Local
!--------------------------------------------------------------
! 1D array used for the input and output of the routine boulac1D
real z1D(kms:kme) ! vertical coordinates (faces of the grid)
real dz1D(kms:kme) ! vertical resolution
real u1D(kms:kme) ! wind speed in the x directions
real v1D(kms:kme) ! wind speed in the y directions
real th1D(kms:kme) ! potential temperature
real q1D(kms:kme) ! moisture
real qc1D(kms:kme) ! liquid water
real rho1D(kms:kme) ! air density
real rhoz1D(kms:kme) ! air density at the faces
real tke1D(kms:kme) ! Turbulent Kinetic Energy
real diss1D(kms:kme) ! Dissipation Rate
real tpe1D(kms:kme)
real th01D(kms:kme) ! reference potential temperature
real exch1D(kms:kme) ! exch
real sf1D(kms:kme) ! surface of the grid cells
real vl1D(kms:kme) ! volume of the grid cells
real a_u1D(kms:kme) ! Implicit component of the momentum sources or sinks in the X-direction
real a_v1D(kms:kme) ! Implicit component of the momentum sources or sinks in the Y-direction
real a_t1D(kms:kme) ! Implicit component of the heat sources or sinks
real a_q1D(kms:kme) ! Implicit component of the moisture sources or sinks
real a_qc1D(kms:kme) ! Implicit component of the liquid water sources or sinks
real b_u1D(kms:kme) ! Explicit component of the momentum sources or sinks in the X-direction
real b_v1D(kms:kme) ! Explicit component of the momentum sources or sinks in the Y-direction
real b_t1D(kms:kme) ! Explicit component of the heat sources or sinks
real b_q1D(kms:kme) ! Explicit component of the moisture sources or sinks
real b_qc1D(kms:kme) ! Explicit component of the liquid water sources or sinks
real b_e1D(kms:kme) ! Explicit component of the moisture sources or sinks
real a_e1D(kms:kme) ! Explicit component of the moisture sources or sinks
real b_d1D(kms:kme) ! Explicit component of the moisture sources or sinks
real a_d1D(kms:kme) ! Explicit component of the moisture sources or sinks
real b_tpe1D(kms:kme) ! Explicit component of the moisture sources or sinks
real a_tpe1D(kms:kme) ! Explicit component of the moisture sources or sinks
real sh1D(kms:kme) ! shear
real bu1D(kms:kme) ! buoyancy
real wu1D(kms:kme) ! turbulent flux of momentum (x component)
real wv1D(kms:kme) ! turbulent flux of momentum (y component)
real wt1D(kms:kme) ! turbulent flux of temperature
real wq1D(kms:kme) ! turbulent flux of water vapor
real wqc1D(kms:kme) ! turbulent flux of liquid water
integer iz_pbl(ims:ime,jms:jme)
real pr1D(kms:kme)
real raten1D(kms:kme)
real p8w1D(kms:kme)
real pi_1D(kms:kme)
! local added only for diagnostic output
real bu(ims:ime,kms:kme,jms:jme) ! buoyancy term in TKE
real sh(ims:ime,kms:kme,jms:jme) ! shear term in TKE
real wrk(ims:ime) ! working array
integer ix,iy,iz,id
real ufrac_int ! urban fraction
real psim1D,psih1D,znt1D,br1D,sflux
integer xtime
!
!
!here I fix the value of the reference state equal to the value of the potnetial temperature
! the only use of this variable in the code is to compute the paramter BETA = g/th0
! I fix it to 300K.
rublten=0.
rvblten=0.
rthblten=0.
rqvblten=0.
rqcblten=0.
do ix=its,ite
do iy=jts,jte
do iz=kts,kte
th_0(ix,iz,iy)=300.
enddo
enddo
enddo
z1D=0.
dz1D=0.
u1D =0.
v1D =0.
th1D=0.
q1D=0.
rho1D=0.
rhoz1D=0.
tke1D =0.
diss1D=0.
tpe1D=0.
raten1D=0.
pi_1D=0.
p8w1D=0.
th01D =0.
exch1D=0.
sf1D =1.
vl1D =1.
a_u1D =0.
a_v1D =0.
a_t1D =0.
a_q1D =0.
a_qc1D =0.
b_u1D =0.
b_v1D =0.
b_t1D =0.
b_q1D =0.
b_qc1D =0.
b_e1D=0.
a_e1D=0.
b_d1D=0.
a_d1D=0.
b_tpe1D=0.
a_tpe1D=0.
sh1D =0.
bu1D =0.
wu1D =0.
wv1D =0.
wt1D =0.
wq1D =0.
iz_pbl=0
psim1D=0.
psih1D=0.
do ix=its,ite
do iy=jts,jte
z1d(kts)=0.
br1D=br(ix,iy)
sflux=hfx(ix,iy)/rho(ix,1,iy)/cp+qfx(ix,iy)/rho(ix,1,iy)*(461./287.-1)*th_phy(ix,1,iy)
znt1D=znt(ix,iy)
psim1D=psim(ix,iy)
psih1D=psih(ix,iy)
do iz= kts,kte
if(xtime.le.2)then
tke_pbl(ix,iz,iy)=temin
diss_pbl(ix,iz,iy)=dmin
tpe_pbl(ix,iz,iy)=tpemin
tke_adv(ix,iz,iy)=temin
diss_adv(ix,iz,iy)=dmin
tpe_adv(ix,iz,iy)=tpemin
pr_pbl(ix,iz,iy)=1.
endif
u1D(iz)=u_phy(ix,iz,iy)
v1D(iz)=v_phy(ix,iz,iy)
th1D(iz)=th_phy(ix,iz,iy)
q1D(iz)=qv_curr(ix,iz,iy)
qc1D(iz)=qc_curr(ix,iz,iy)
rho1D(iz)=rho(ix,iz,iy)
th01D(iz)=th_0(ix,iz,iy)
dz1D(iz)=dz8w(ix,iz,iy)
pi_1D(iz)=pi_phy(ix,iz,iy)
raten1D(iz)=rthraten(ix,iz,iy)
p8w1D(iz)=p8w(ix,iz,iy)
pr1D(iz)=pr_pbl(ix,iz,iy)
z1D(iz+1)=z1D(iz)+dz1D(iz)
if(bl_keps_adv==1)then
tke_pbl(ix,iz,iy)=tke_adv(ix,iz,iy)
diss_pbl(ix,iz,iy)=diss_adv(ix,iz,iy)
tpe_pbl(ix,iz,iy)=tpe_adv(ix,iz,iy)
endif
tke1D(iz)=min(temax,max(temin,tke_pbl(ix,iz,iy)))
diss1D(iz)=min(dmax,max(dmin,diss_pbl(ix,iz,iy)))
tpe1D(iz)=min(tpemax,max(tpemin,tpe_pbl(ix,iz,iy)))
enddo
rhoz1D(kts)=rho1D(kts)
do iz=kts+1,kte
rhoz1D(iz)=(rho1D(iz)*dz1D(iz-1)+rho1D(iz-1)*dz1D(iz))/(dz1D(iz-1)+dz1D(iz))
enddo
rhoz1D(kte+1)=rho1D(kte)
do iz=kts,kte
vl1D(iz)=1.
sf1D(iz)=1.
enddo
ufrac_int=0.
sf1D(kte+1)=1.
! compute the pbl_height
call GET_PBLH(kts,kte,pblh(ix,iy),th1D,tke1D,z1D,dz1D,q1D,iz_pbl(ix,iy))
!call pbl_height(kms,kme,kts,kte,dz1D,z1D,th1D,q1D,pblh(ix,iy),iz_pbl(ix,iy))
! estimate the tendencies
do iz=kts,kte
a_t1D(iz)=0.
b_t1D(iz)=0.
a_u1D(iz)=0.
b_u1D(iz)=0.
a_v1D(iz)=0.
b_v1D(iz)=0.
a_q1D(iz)=0.
b_q1D(iz)=0.
b_e1D(iz)=0.
a_e1D(iz)=0.
b_d1D(iz)=0.
a_d1D(iz)=0.
b_tpe1D(iz)=0.
a_tpe1D(iz)=0.
enddo
b_t1D(1)=hfx(ix,iy)/dz1D(1)/rho1D(1)/cp
b_q1D(1)=qfx(ix,iy)/dz1D(1)/rho1D(1)
a_u1D(1)=(-ustar(ix,iy)*ustar(ix,iy)/dz1D(1)/((max(1.E-10,abs(u1D(1)))**2.+max(1.E-10,abs(v1D(1)))**2.)**.5))
a_v1D(1)=(-ustar(ix,iy)*ustar(ix,iy)/dz1D(1)/((max(1.E-10,abs(u1D(1)))**2.+max(1.E-10,abs(v1D(1)))**2.)**.5))
if(xtime.gt.2)then
call keps1D(mol(ix,iy),tsk(ix,iy),xtime,ix,iy,ids,ide,jds,jde,kms,kme,kts,kte,dz1D,z1D,dt, &
u1D,v1D,th1D,pi_1D,raten1D,p8w1D, &
rho1D,rhoz1D,q1D,qc1D,th01D,tke1D,diss1D,tpe1D,pr1D, &
a_u1D,b_u1D,a_v1D,b_v1D,a_t1D,b_t1D,a_q1D,b_q1D,a_qc1D,b_qc1D, &
b_e1D,a_e1D,b_d1D,a_d1D,b_tpe1D,a_tpe1D,psim1D,psih1D, &
ustar(ix,iy),hfx(ix,iy),qfx(ix,iy),sflux,cp,g, &
sf1D,vl1D,exch1D,sh1D,bu1D,br1D,znt1D, &
pblh(ix,iy),iz_pbl(ix,iy),ufrac_int)
endif
do iz= kts,kte
tke_pbl(ix,iz,iy)=tke1D(iz)
diss_pbl(ix,iz,iy)=diss1D(iz)
tpe_pbl(ix,iz,iy)=tpe1D(iz)
if(bl_keps_adv==1)then
tke_adv(ix,iz,iy)=(tke_pbl(ix,iz,iy))
diss_adv(ix,iz,iy)=diss_pbl(ix,iz,iy)
tpe_adv(ix,iz,iy)=tpe_pbl(ix,iz,iy)
endif
pr_pbl(ix,iz,iy)=pr1D(iz)
bu(ix,iz,iy)=bu1D(iz)
exch_h(ix,iz,iy)=exch1D(iz)/pr1D(iz)
exch_m(ix,iz,iy)=exch1D(iz)
enddo
! compute the tendencies
do iz= kts,kte
rthblten(ix,iz,iy)=rthblten(ix,iz,iy)+(th1D(iz)-th_phy(ix,iz,iy))/dt
rqvblten(ix,iz,iy)=rqvblten(ix,iz,iy)+(q1D(iz)-qv_curr(ix,iz,iy))/dt
rqcblten(ix,iz,iy)=rqcblten(ix,iz,iy)+(qc1D(iz)-qc_curr(ix,iz,iy))/dt
rublten(ix,iz,iy)=rublten(ix,iz,iy)+(u1D(iz)-u_phy(ix,iz,iy))/dt
rvblten(ix,iz,iy)=rvblten(ix,iz,iy)+(v1D(iz)-v_phy(ix,iz,iy))/dt
enddo
! endif
wt(ix,kts,iy)=hfx(ix,iy)/rho1D(kts)/cp-rho1D(kts)*(th1D(kts)-th_phy(ix,kts,iy))/dt*dz1D(kts)
wu(ix,kts,iy)=-ustar(ix,iy)*ustar(ix,iy)-rho1D(kts)*(u1D(kts)-u_phy(ix,kts,iy))/dt*dz1D(kts)
wv(ix,kts,iy)=-ustar(ix,iy)*ustar(ix,iy)-rho1D(kts)*(u1D(kts)-u_phy(ix,kts,iy))/dt*dz1D(kts)
do iz=kts+1,kte
wt(ix,iz,iy)=wt(ix,iz-1,iy)-rho1D(iz)*(th1D(iz)-th_phy(ix,iz,iy))/dt*dz1D(iz)
wu(ix,iz,iy)=wu(ix,iz-1,iy)-rho1D(iz)*(u1D(iz)-u_phy(ix,iz,iy))/dt*dz1D(iz)
wv(ix,iz,iy)=wv(ix,iz-1,iy)-rho1D(iz)*(v1D(iz)-v_phy(ix,iz,iy))/dt*dz1D(iz)
enddo
enddo ! iy
enddo ! ix
return
end subroutine keps
! ===6=8===============================================================72
! ===6=8===============================================================72
subroutine keps1D(mol,tsk,xtime,ix,iy,ids,ide,jds,jde,kms,kme,kts,kte,dz,z,dt, &
u,v,th,pi1d,raten,p8w1D,rho,rhoz,qa,qc,th0,te,d,tpe,Pra, &
a_u,b_u,a_v,b_v,a_t,b_t,a_q,b_q,a_qc,b_qc,b_e,a_e, &
b_d,a_d,a_tpe,b_tpe, &
psim,psih,ustar,hfx,qfx,sflux,cp,g, &
sf,vl,exch,sh,bu,b_ric,z0,pblh,iz_pbl,ufrac_int)
! ----------------------------------------------------------------------
! 1D resolution of TKE and dissipation rate following k epsilon turbulent closure
! ----------------------------------------------------------------------
implicit none
integer iz,ix,iy,xtime
integer kms,kme,kts,kte
integer ids,ide,jds,jde
real z(kms:kme) ! Altitude above the ground of the cell interfaces.
real dz(kms:kme) ! vertical resolution
real dt,time ! Time step
real pblh
real tsk,mol
real u(kms:kme) ! Wind speed in the x direction
real v(kms:kme) ! Wind speed in the y direction
real th(kms:kme) ! Potential temperature
real rho(kms:kme) ! Air density
real rhoz(kms:kme) !air density at the faces of the cell
real qa(kms:kme) ! air humidity
real qc(kms:kme) ! air humidity
real th0(kms:kme) ! Reference potential temperature
real te(kms:kme) ! turbulent kinetic energy
real d(kms:kme) ! dissipation rate
real tpe(kms:kme)
real pi1d(kms:kme)
real raten(kms:kme)
real p8w1D(kms:kme)
real a_u(kms:kme)
real b_u(kms:kme)
real a_v(kms:kme)
real b_v(kms:kme)
real a_t(kms:kme)
real b_t(kms:kme)
real a_q(kms:kme)
real b_q(kms:kme)
real a_qc(kms:kme)
real b_qc(kms:kme)
real a_e(kms:kme)
real b_e(kms:kme)
real a_d(kms:kme)
real b_d(kms:kme)
real a_tpe(kms:kme)
real b_tpe(kms:kme)
real ufrac_int
real ustar ! ustar
real hfx ! sensbile heat flux
real qfx ! kinematic latent heat flux
real g ! gravity
real cp !
real sf(kms:kme) ! surface of the urban grid cells
real vl(kms:kme) ! volume of the urban grid cells
real exch(kms:kme) ! turbulent diffusion coefficients (defined at the faces)
real sh(kms:kme) ! shear term in TKE eqn.
real bu(kms:kme) ! buoyancy term in TKE eqn.
real b_ric,z0,sflux
integer iz_pbl
real we(kms:kme)
real wd(kms:kme)
real wtpe(kms:kme)
real wu(kms:kme)
real wv(kms:kme)
real wt(kms:kme)
real wq(kms:kme)
real wqc(kms:kme)
real Ri(kms:kme)
real Pra(kms:kme),coeff(kms:kme)
real Pra_st(kms:kme)
real exch_h(kms:kme)
real nsquare(kms:kme)
real dls(kms:kme)
real psim,psih
real phim,phih,phieps
real dtmdz(kms:kme)
call shear(kms,kme,kts,kte,u,v,dz,sh,sf)
call c_nsquare(Pra,ustar,sflux,z0,tsk,mol,pi1d,g,kms,kme,kts,kte,th,th0,dz,nsquare,dtmdz)
call surface_bl_pra_ri(kms,kme,kts,kte,dz,z,rho,g,cp,z0,sflux,raten,pi1d,p8w1D, &
th,qa,sh,nsquare,Ri,b_ric,psim,psih,pblh,ustar, &
iz_pbl,Pra,Pra_st,phim,phih,phieps,dtmdz(kts))
call buoy(g,kms,kme,kts,kte,th,th0,dz,bu,te,tpe,Pra)
call cdtur_ke(kms,kme,kts,kte,te,d,z,dz,exch,exch_h,Pra_st)
call length_bougeault(g,kms,kme,kts,kte,z,dz,te,th,th0,dls)
!solve analytically the equation for tke and dissipation,following Zonato et
!al.2022
call thetaphi_alphabeta(kms,kme,kts,kte,z0,dz,sh,bu,te,d,dt,Pra,dls,ustar,phieps,phim,Ri,nsquare,a_d)
! solve diffusion equation for tke
call diff(kms,kme,kts,kte,2,1,dt,te,rho,rhoz,exch,a_e,b_e,sf,vl,dz,we)
! solve diffusion equation for dissipation rate
call diff(kms,kme,kts,kte,2,1,dt,d,rho,rhoz,exch/sigma_eps,a_d,b_d,sf,vl,dz,wd)
!calculation of the terms for turbulent potential energy equation and addition
!of the countergradient term to tke
call terms_tpe(dtmdz,g,kms,kme,kts,kte,th0,te,d,tpe,a_tpe,b_tpe,Pra,coeff)
call calc_countergradient(dz,g,kms,kme,kts,kte,th0,te,d,tpe,Pra,b_t)
!solve diffusion equation for turbulent potential energy
call diff(kms,kme,kts,kte,1,1,dt,tpe,rho,rhoz,exch,a_tpe,b_tpe,sf,vl,dz,wtpe)
! solve diffusion equation for momentum x component
call diff(kms,kme,kts,kte,1,1,dt,u,rho,rhoz,exch,a_u,b_u,sf,vl,dz,wu)
! solve diffusion equation for momentum y component
call diff(kms,kme,kts,kte,1,1,dt,v,rho,rhoz,exch,a_v,b_v,sf,vl,dz,wv)
! solve diffusion equation for potential temperature
call diff(kms,kme,kts,kte,1,1,dt,th,rho,rhoz,exch_h,a_t,b_t,sf,vl,dz,wt)
! solve diffusion equation for water vapor mixing ratio
call diff(kms,kme,kts,kte,1,1,dt,qa,rho,rhoz,exch_h,a_q,b_q,sf,vl,dz,wq)
! solve diffusion equation for liquid water mixing ratio
call diff(kms,kme,kts,kte,1,1,dt,qc,rho,rhoz,exch_h,a_qc,b_qc,sf,vl,dz,wqc)
do iz=kts,kte
te(iz)=min(max(temin,te(iz)),temax)
d(iz)=min(max(dmin,d(iz)),dmax)
tpe(iz)=min(max(tpemin,tpe(iz)),tpemax)
enddo
return
end subroutine keps1D
! ===6=8===============================================================72
! ===6=8===============================================================72
subroutine cdtur_ke(kms,kme,kts,kte,te,d,z,dz,exch,exch_h,Pra_st)
! compute turbulent coefficients
implicit none
integer iz,kms,kme,kts,kte
real te(kms:kme),exch(kms:kme),exch_h(kms:kme),d(kms:kme)
real dz(kms:kme),z(kms:kme),Pra_st(kms:kme)
real fact
exch(kts)=0.
exch_h(kts)=0.
do iz=kts+1,kte
exch(iz)=(c_mu*te(iz-1)**2/d(iz-1)*dz(iz-1)+c_mu*te(iz)**2/d(iz)*dz(iz))/(dz(iz)+dz(iz-1))
exch(iz)=max(exch(iz),0.09)
exch_h(iz)=exch(iz)/Pra_st(iz)
enddo
exch(kte+1)=exch(kte)
exch_h(kte+1)=exch_h(kte)
return
end subroutine cdtur_ke
! ===6=8===============================================================72
! ===6=8===============================================================72
subroutine diff(kms,kme,kts,kte,iz1,izf,dt,co,rho,rhoz,cd,aa,bb,sf,vl,dz,fc)
!------------------------------------------------------------------------
! Calculation of the diffusion in 1D
!------------------------------------------------------------------------
! - Input:
! nz : number of points
! iz1 : first calculated point
! co : concentration of the variable of interest
! dz : vertical levels
! cd : diffusion coefficients
! dtext : external time step
! rho : density of the air at the center
! rhoz : density of the air at the face
! itest : if itest eq 1 then update co, else store in a flux array
! - Output:
! co :concentration of the variable of interest
! - Internal:
! cddz : constant terms in the equations
! dt : diffusion time step
! nt : number of the diffusion time steps
! cstab : ratio of the stability condition for the time step
!--------------------------------------------------------------
implicit none
integer iz,iz1,izf
integer kms,kme,kts,kte
real dt,dzv
real co(kms:kme),cd(kms:kme),dz(kms:kme)
real rho(kms:kme),rhoz(kms:kme)
real cddz(kms:kme+1),fc(kms:kme),df(kms:kme)
real a(kms:kme,3),c(kms:kme)
real sf(kms:kme),vl(kms:kme)
real aa(kms:kme),bb(kms:kme)
! Compute cddz=2*cd/dz
cddz(kts)=sf(kts)*rhoz(kts)*cd(kts)/dz(kts)
do iz=kts+1,kte
cddz(iz)=2.*sf(iz)*rhoz(iz)*cd(iz)/(dz(iz)+dz(iz-1))
enddo
cddz(kte+1)=sf(kte+1)*rhoz(kte+1)*cd(kte+1)/dz(kte)
do iz=kts,iz1-1
a(iz,1)=0.
a(iz,2)=1.
a(iz,3)=0.
c(iz)=co(iz)
enddo
do iz=iz1,kte-izf
dzv=vl(iz)*dz(iz)
a(iz,1)=-cddz(iz)*dt/dzv/rho(iz)
a(iz,2)=1+dt*(cddz(iz)+cddz(iz+1))/dzv/rho(iz)-aa(iz)*dt
a(iz,3)=-cddz(iz+1)*dt/dzv/rho(iz)
c(iz)=co(iz)+bb(iz)*dt
enddo
do iz=kte-(izf-1),kte
a(iz,1)=0.
a(iz,2)=1
a(iz,3)=0.
c(iz)=co(iz)
enddo
call invert (kms,kme,kts,kte,a,c,co)
do iz=kts,iz1
fc(iz)=0.
enddo
do iz=iz1+1,kte
fc(iz)=-(cddz(iz)*(co(iz)-co(iz-1)))/rho(iz)
enddo
return
end subroutine diff
! ===6=8===============================================================72
! ===6=8===============================================================72
subroutine buoy(g,kms,kme,kts,kte,th,th0,dz,bu,te,tpe,Pra)
implicit none
integer kms,kme,kts,kte,iz
real dtdz1,dtdz2,g,dtmdz
real th(kms:kme),dz(kms:kme),bu(kms:kme),sf(kms:kme)
real th0(kms:kme),phi_bu(kms:kme)
real te(kms:kme),tpe(kms:kme),Pra(kms:kme)
bu(kts)=0.
do iz=kts+1,kte-1
phi_bu(iz)=g/th0(iz)*tpe(iz)/te(iz)*Pra(iz)*c_theta/c_mu
dtdz1=2.*(th(iz)-th(iz-1))/(dz(iz-1)+dz(iz))
dtdz2=2.*(th(iz+1)-th(iz))/(dz(iz+1)+dz(iz))
dtmdz=0.5*(dtdz1+dtdz2)
bu(iz)=-dtmdz*g/th0(iz)+(g/th0(iz))*phi_bu(iz)
enddo
bu(kte)=0.
return
end subroutine buoy
! ===6=8===============================================================72
! ===6=8===============================================================72
subroutine c_nsquare(Pra,ustar,sflux,z0,tsk,mol,pi,g,kms,kme,kts,kte,th,th0,dz,nsquare,dtmdz)
implicit none
integer kms,kme,kts,kte,iz
real thetaz0,sflux,ustar,z0,dtdz1,dtdz2,g,tsk,mol,th0(kms:kme)
real th(kms:kme),dz(kms:kme),nsquare(kms:kme),sf(kms:kme)
real te(kms:kme),d(kms:kme),dtmdz(kms:kme),Pra(kms:kme)
real pi(kms:kme)
dtmdz(kts)=(-sflux/ustar)/vk/(dz(kts)/2.-z0)
nsquare(kts)=-g/th0(kts)*dtmdz(kts)
do iz=kts+1,kte-1
dtdz1=2.*(th(iz)-th(iz-1))/(dz(iz-1)+dz(iz))
dtdz2=2.*(th(iz+1)-th(iz))/(dz(iz+1)+dz(iz))
dtmdz(iz)=0.5*(dtdz1+dtdz2)
nsquare(iz)=-dtmdz(iz)*g/th0(iz)
enddo
nsquare(kte)=0.
return
end subroutine c_nsquare
! ===6=8===============================================================72
! ===6=8===============================================================72
subroutine shear(kms,kme,kts,kte,u,v,dz,sh,sf)
! compute shear term
implicit none
integer kms,kme,kts,kte,iz,ix,iy
real dudz1,dudz2,dvdz1,dvdz2,dumdz
real u(kms:kme),v(kms:kme),dz(kms:kme),sh(kms:kme),sf(kms:kme)
real u1,u2,v1,v2,ufrac_int
sh(kts)=0.
do iz=kts+1,kte-1
u2=(dz(iz+1)*u(iz)+dz(iz)*u(iz+1))/(dz(iz)+dz(iz+1))
u1=(dz(iz)*u(iz-1)+dz(iz-1)*u(iz))/(dz(iz-1)+dz(iz))
v2=(dz(iz+1)*v(iz)+dz(iz)*v(iz+1))/(dz(iz)+dz(iz+1))
v1=(dz(iz)*v(iz-1)+dz(iz-1)*v(iz))/(dz(iz-1)+dz(iz))
dumdz=((u2-u1)/dz(iz))**2.+((v2-v1)/dz(iz))**2.
sh(iz)=dumdz
enddo
sh(kte)=0.
return
end subroutine shear
! ===6=8===============================================================72
! ===6=8===============================================================72
subroutine invert(kms,kme,kts,kte,a,c,x)
implicit none
integer in
integer kts,kte,kms,kme
real a(kms:kme,3),c(kms:kme),x(kms:kme)
do in=kte-1,kts,-1
c(in)=c(in)-a(in,3)*c(in+1)/a(in+1,2)
a(in,2)=a(in,2)-a(in,3)*a(in+1,1)/a(in+1,2)
enddo
do in=kts+1,kte
c(in)=c(in)-a(in,1)*c(in-1)/a(in-1,2)
enddo
do in=kts,kte
x(in)=c(in)/a(in,2)
enddo
return
end subroutine invert
!######################################################################
subroutine pbl_height(kms,kme,kts,kte,dz,z,th,q,pblh,iz_pbl)
! this routine computes the PBL height
! with an approach similar to MYNN
implicit none
integer kms,kme,kts,kte,iz
real z(kms:kme),dz(kms:kme),th(kms:kme),q(kms:kme)
real pblh
!Local
real thv(kms:kme),zc(kms:kme)
real thsfc
integer iz_pbl
! compute the height of the center of the grid cells
do iz=kts,kte
zc(iz)=z(iz)+dz(iz)/2.
enddo
! compute the virtual potential temperature
do iz=kts,kte
thv(iz)=th(iz)*(1.+0.61*q(iz))
enddo
! now compute the PBL height
pblh=0.
thsfc=thv(kts)+0.5
iz_pbl=1
do iz=kts+1,kte
if(pblh.eq.0.and.thv(iz).gt.thsfc)then
pblh=zc(iz-1)+(thsfc-thv(iz-1))/(max(0.01,thv(iz)-thv(iz-1)))*(zc(iz)-zc(iz-1))
endif
enddo
do iz=kts,kte
if(zc(iz).le.pblh)then
iz_pbl=iz_pbl+1
endif
enddo
return
end subroutine pbl_height
! ===6=8===============================================================72
! ===6=8===============================================================72
SUBROUTINE GET_PBLH(KTS,KTE,zi,theta1D,tke1D,zw1D,dz1D,q1D,iz_pbl)
! Copied from MYNN PBL
!---------------------------------------------------------------
! NOTES ON THE PBLH FORMULATION
!
!The 1.5-theta-increase method defines PBL heights as the level at
!which the potential temperature first exceeds the minimum potential
!temperature within the boundary layer by 1.5 K. When applied to
!observed temperatures, this method has been shown to produce PBL-
!height estimates that are unbiased relative to profiler-based
!estimates (Nielsen-Gammon et al. 2008). However, their study did not
!include LLJs. Banta and Pichugina (2008) show that a TKE-based
!threshold is a good estimate of the PBL height in LLJs. Therefore,
!a hybrid definition is implemented that uses both methods, weighting
!the TKE-method more during stable conditions (PBLH < 400 m).
!A variable tke threshold (TKEeps) is used since no hard-wired
!value could be found to work best in all conditions.
!---------------------------------------------------------------
INTEGER,INTENT(IN) :: KTS,KTE
REAL, INTENT(OUT) :: zi
REAL, DIMENSION(KTS:KTE), INTENT(IN) :: tke1D, dz1D,theta1D,q1D
REAL, DIMENSION(KTS:KTE) :: thetav1D
REAL, DIMENSION(KTS:KTE+1), INTENT(IN) :: zw1D
!LOCAL VARS
REAL :: PBLH_TKE,tke,tkem1,wt,maxtke,TKEeps,minthv
REAL :: delt_thv !delta theta-v; dependent on land/sea point
REAL, PARAMETER :: sbl_lim = 200. !Theta-v PBL lower limit of trust (m).
REAL, PARAMETER :: sbl_damp = 400. !Damping range for averaging with TKE-based PBLH (m).
INTEGER :: I,J,K,kthv,ktke,iz_pbl
do iz=kts,kte
thetav1D(iz)=theta1D(iz)*(1.+0.61*q1D(iz))
enddo
!FIND MAX TKE AND MIN THETAV IN THE LOWEST 500 M
k = kts+1
kthv = 1
ktke = 1
maxtke = 0.
minthv = 9.E9
DO WHILE (zw1D(k) .LE. 500.)
tke =MAX(tke1D(k),0.) ! maximum QKE
IF (maxtke < ttke) then
maxtke = ttke
ktke = k
ENDIF
IF (minthv > thetav1D(k)) then
minthv = thetav1D(k)
kthv = k
ENDIF
k = k+1
ENDDO
TKEeps = maxtke/20.
TKEeps = MAX(TKEeps,0.025/2.)
TKEeps = MIN(TKEeps,0.25/2.)
!FIND THETAV-BASED PBLH (BEST FOR DAYTIME).
zi=0.
k = kthv+1
delt_thv = 1.5
DO WHILE (zi .EQ. 0.)
IF (thetav1D(k) .GE. (minthv + delt_thv))THEN
zi = zw1D(k) - dz1D(k-1)* &
& MIN((thetav1D(k)-(minthv+delt_thv))/MAX(thetav1D(k)-thetav1D(k-1),1E-6),1.0)
ENDIF
k = k+1
IF (k .EQ. kte-1) zi = zw1D(kts+1) !EXIT SAFEGUARD
ENDDO
PBLH_TKE=0.
k = ktke+1
DO WHILE (PBLH_TKE .EQ. 0.)
tke =MAX(tke1D(k)/2.,0.) ! maximum TKE
tkem1=MAX(tke1D(k-1)/2.,0.)
IF (tke .LE. TKEeps) THEN
PBLH_TKE = zw1D(k) - dz1D(k-1)* &
& MIN((TKEeps-tke)/MAX(tkem1-tke, 1E-6), 1.0)
!IN CASE OF NEAR ZERO TKE, SET PBLH = LOWEST LEVEL.
PBLH_TKE = MAX(PBLH_TKE,zw1D(kts+1))
!print *,"PBLH_TKE:",i,j,PBLH_TKE, Qke1D(k)/2., zw1D(kts+1)
ENDIF
k = k+1
IF (k .EQ. kte-1) PBLH_TKE = zw1D(kts+1) !EXIT SAFEGUARD
ENDDO
!BLEND THE TWO PBLH TYPES HERE:
wt=.5*TANH((zi - sbl_lim)/sbl_damp) + .5
zi=PBLH_TKE*(1.-wt) + zi*wt
iz_pbl=1
do k=kts,kte
if(zw1D(k).le.zi)then
iz_pbl=iz_pbl+1
endif
enddo
END SUBROUTINE GET_PBLH
! ===6=8===============================================================72
! ===6=8===============================================================72
subroutine thetaphi_alphabeta(kms,kme,kts,kte,z0,dz,sh,bu,te,d,dt,Pra,dls,ustar,phieps,phim,Ri,nsquare,a_d)
implicit none
integer iz,kms,kme,kts,kte
real te(kms:kme) ! turbulent kinetic energy
real d(kms:kme) ! dissipation rate
real a_d(kms:kme)
real dt,z0 ! Time step
real sh(kms:kme) ! shear term in TKE eqn.
real bu(kms:kme) ! buoyancy term in TKE eqn.
real A(kms:kme)
real B(kms:kme)
real theta(kms:kme)
real phi(kms:kme)
real theta_new(kms:kme)
real phi_new(kms:kme)
real C(kms:kme)
real alpha
real beta
real F
real dz(kms:kme)
real Pra(kms:kme)
real dls(kms:kme)
real Ri(kms:kme)
real nsquare(kms:kme)
real ustar,phieps,phim
alpha=1.
beta=-1./c2
F=c2-1.
do iz=kts+1,kte
if(abs(te(iz))/abs(d(iz)).gt.1E4.and.d(iz).le.1E-6)then
d(iz)=1/1.4*te(iz)**(3./2.)/dls(iz)
endif
theta(iz)=max(temin,te(iz))/max(dmin,d(iz))
phi(iz)=max(dmin,d(iz))**beta*max(temin,te(iz))**alpha
C(iz)=(c_mu*((c1-1.)*sh(iz)+((c3-1.)*bu(iz))/Pra(iz)))
A(iz)=c_mu*(sh(iz)+bu(iz)/Pra(iz))
B(iz)=c_mu*(c1*sh(iz)+c3*bu(iz)/Pra(iz))
if(C(iz).lt.0.)then
C(iz)=-C(iz)
theta_new(iz)=(c2-1.)/(C(iz)*theta(iz))+((c2-1.)**(1./2.)*(C(iz)*theta(iz)**2-c2+1))/(C(iz)*theta(iz)*((c2-1.)**(1./2.)+C(iz)**(1./2.)*theta(iz)*tanh(C(iz)**(1./2.)*dt*(c2-1)**(1./2.))))
elseif(C(iz).eq.0.)then
theta_new(iz)=theta(iz)+F*dt;
else
theta_new(iz)=((F/C(iz))**(0.5)*(tanh(dt*(C(iz)*F)**(0.5)) +theta(iz)*(C(iz)/F)**(0.5)))/(theta(iz)*tanh(dt*(C(iz)*F)**(0.5))*(C(iz)/F)**(0.5) + 1.)
endif
theta_new(iz)=max(1.,theta_new(iz))
phi_new(iz)=(phi(iz)*exp(theta_new(iz)*dt*(alpha*A(iz)+beta*B(iz))))
te(iz)=((phi_new(iz)*theta_new(iz)**(beta))**(1./(alpha+beta)))
d(iz)=(te(iz)/theta_new(iz))
enddo
do iz=kts,kte
if(Ri(iz).gt.0.)then
a_d(iz)=a_d(iz)+c4*min(1.,sqrt(Ri(iz)/c5))*sqrt(max(0.,-nsquare(iz)))
endif
enddo
d(kts)=(ustar**3./vk/(dz(1)/2.-z0))*phieps*(1.+a_d(kts)*d(kts)*dt)
te(kts)=(ustar**2.)/sqrt(c_mu)*sqrt(phieps/phim)
return
end subroutine thetaphi_alphabeta
! ===6=8===============================================================72
! ===6=8===============================================================72
subroutine length_bougeault(g,kms,kme,kts,kte,z,dz,te,th,th0,dls)
! compute the length scales up and down
implicit none
integer kms,kme,kts,kte,iz,izz,ix,iy
real dzt,zup,beta,zup_inf,bbb,tl,zdo,zdo_sup,zzz,g
real te(kms:kme),dlu(kms:kme),dld(kms:kme),dz(kms:kme)
real z(kms:kme),th(kms:kme),th0(kms:kme),dls(kms:kme)
do iz=kts,kte
zup=0.
dlu(iz)=z(kte+1)-z(iz)-dz(iz)/2.
zzz=0.
zup_inf=0.
beta=g/th0(iz) !Buoyancy coefficient
do izz=iz,kte-1
dzt=(dz(izz+1)+dz(izz))/2.
zup=zup-beta*th(iz)*dzt
zup=zup+beta*(th(izz+1)+th(izz))*dzt/2.
zzz=zzz+dzt
if(te(iz).lt.zup.and.te(iz).ge.zup_inf)then
bbb=(th(izz+1)-th(izz))/dzt
if(bbb.ne.0)then
tl=(-beta*(th(izz)-th(iz))+sqrt( max(0.,(beta*(th(izz)-th(iz)))**2.+2.*bbb*beta*(te(iz)-zup_inf))))/bbb/beta
else
if(th(izz).ne.th(iz))then
tl=(te(iz)-zup_inf)/(beta*(th(izz)-th(iz)))
else
tl=0.
endif
endif
dlu(iz)=zzz-dzt+tl
endif
zup_inf=zup
enddo
zdo=0.
zdo_sup=0.
dld(iz)=z(iz)+dz(iz)/2.
zzz=0.
do izz=iz,kts+1,-1
dzt=(dz(izz-1)+dz(izz))/2.
zdo=zdo+beta*th(iz)*dzt
zdo=zdo-beta*(th(izz-1)+th(izz))*dzt/2.
zzz=zzz+dzt
if(te(iz).lt.zdo.and.te(iz).ge.zdo_sup)then
bbb=(th(izz)-th(izz-1))/dzt
if(bbb.ne.0.)then
tl=(beta*(th(izz)-th(iz))+sqrt( max(0.,(beta*(th(izz)-th(iz)))**2.+2.*bbb*beta*(te(iz)-zdo_sup))))/bbb/beta
else
if(th(izz).ne.th(iz))then
tl=(te(iz)-zdo_sup)/(beta*(th(izz)-th(iz)))
else